Bicomplexes

Assume we have an array of $k$-modules

$$\xymatrix{ \vdots \ar[d] & \vdots \ar[d] & \vdots \ar[d] & \... ..._{10} \ar[l]_{d^h} & C_{20} \ar[l]_{d^h} & \hdots \ar[l] \\ }$$

We call it a bicomplex of $k$-modules if the maps $d^v$ and $d^h$, called vertical and horizontal differential, satisfy
$$\begin{align*} d^v\circ d^v &= 0\\ d^h\circ d^h &= 0\\ d^h\circ d^v + d^v\circ d^h &= 0\\ \end{align*}$$
For a bicomplex $C_{\bullet\bullet}$ we define a total complex as

$$\Tot(C_{\bullet\bullet})_n:=\bigoplus_{p+q=n}C_{pq},\quad d:=d^h+d^v$$

After taking homology with respect to the vertical differential we obtain a complex

$$\hdots \leftarrow \rH^{v}_{(p-1),\bullet} \leftarrow \rH^v_{p, \bullet} \leftarrow \rH^v_{(p+1), \bullet} \leftarrow \hdots$$

with the differential induced on homology by horizontal differential in the bicomplex. Now we can take homology of this complex and obtain

$$E_{pq}^2:=\rH^h_q(\rH^v_{p,\bullet})$$

There is a decomposition of the reduced Hochschild complex

$$\Bar{C}_n(A, A)=A\ox\Bar{A}^{\ox n}=(\Bar{A}\oplus k1)\ox\Bar{A}^{\ox n} =\Bar{A}^{\ox(n+1)}\oplus \Bar{A}^{\ox n}$$

and a map

$$\twobytwo{b}{1-t}{0}{-b'}\: \Bar{A}^{\ox(n+1)}\oplus \Bar{A}^{\ox n} \to \Bar{A}^{\ox n}\oplus \Bar{A}^{\ox (n-1)}$$

which fits in the diagram

$$\xymatrix{ \Bar{C}_n(A, A)\ar[d]_{b} \ar[r]^{\isom}& \Bar{A}... ...ar[r]^{\isom}& \Bar{A}^{\ox n}\oplus \Bar{A}^{\ox (n-1)} \\ }$$

This complex can be thought of as the total complex of a bicomplex

$$\xymatrix{ \vdots \ar[d] & \vdots \ar[d] \\ \Bar{A}^{\ox 3}... ...[l]_{1-t} \ar[d]_{-b'}\\ \Bar{A} & \Bar{A} \ar[l]_{1-t} \\ }$$

Here we see the beginning of the complex computing the homology of the cyclic group with coefficients in a module. This will lead to the cyclic bicomplex.